2D forced-dissipative turbulence

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A simulation of forced-dissipative two-dimensional turbulence. We solve the two-dimensional vorticity equation with stochastic excitation and dissipation in the form of linear drag and hyperviscosity.

using FourierFlows, Printf, Plots

using FourierFlows: parsevalsum
using Random: seed!
using FFTW: irfft

import GeophysicalFlows.TwoDNavierStokes
import GeophysicalFlows.TwoDNavierStokes: energy, enstrophy

Choosing a device: CPU or GPU

dev = CPU()    # Device (CPU/GPU)

Numerical, domain, and simulation parameters

First, we pick some numerical and physical parameters for our model.

 n, L  = 256, 2π             # grid resolution and domain length
 ν, nν = 2e-7, 2             # hyperviscosity coefficient and hyperviscosity order
 μ, nμ = 1e-1, 0             # linear drag coefficient
    dt = 0.005               # timestep
nsteps = 4000                # total number of steps
 nsubs = 20                  # number of steps between each plot

Forcing

We force the vorticity equation with stochastic excitation that is delta-correlated in time and while spatially homogeneously and isotropically correlated. The forcing has a spectrum with power in a ring in wavenumber space of radius $k_f$ and width $\delta k_f$, and it injects energy per unit area and per unit time equal to $\varepsilon$.

forcing_wavenumber = 14.0 * 2π/L   # the central forcing wavenumber for a spectrum that is a ring in wavenumber space
forcing_bandwidth  = 1.5  * 2π/L   # the width of the forcing spectrum
ε = 0.1                            # energy input rate by the forcing

grid = TwoDGrid(dev, n, L)

K = @. sqrt(grid.Krsq)
forcing_spectrum = @. exp(-(K - forcing_wavenumber)^2 / (2 * forcing_bandwidth^2))
@. forcing_spectrum = ifelse(K < 2  * 2π/L, 0, forcing_spectrum)      # no power at low wavenumbers
@. forcing_spectrum = ifelse(K > 20 * 2π/L, 0, forcing_spectrum)      # no power at high wavenumbers
ε0 = parsevalsum(forcing_spectrum .* grid.invKrsq / 2, grid) / (grid.Lx * grid.Ly)
@. forcing_spectrum *= ε / ε0             # normalize forcing to inject energy at rate ε

seed!(1234)

Next we construct function calcF! that computes a forcing realization every timestep

function calcF!(Fh, sol, t, clock, vars, params, grid)
  ξ = ArrayType(dev)(exp.(2π * im * rand(eltype(grid), size(sol))) / sqrt(clock.dt))
  ξ[1, 1] = 0
  @. Fh = ξ * sqrt(forcing_spectrum)

  return nothing
end

Problem setup

We initialize a Problem by providing a set of keyword arguments. The stepper keyword defines the time-stepper to be used.

prob = TwoDNavierStokes.Problem(dev; nx=n, Lx=L, ν=ν, nν=nν, μ=μ, nμ=nμ, dt=dt, stepper="ETDRK4",
                                calcF=calcF!, stochastic=true)

Define some shortcuts for convenience.

sol, clock, vars, params, grid = prob.sol, prob.clock, prob.vars, prob.params, prob.grid

x, y = grid.x, grid.y

First let's see how a forcing realization looks like. Function calcF!() computes the forcing in Fourier space and saves it into variable vars.Fh, so we first need to go back to physical space.

calcF!(vars.Fh, sol, 0.0, clock, vars, params, grid)

heatmap(x, y, irfft(vars.Fh, grid.nx)',
     aspectratio = 1,
               c = :balance,
            clim = (-200, 200),
           xlims = (-L/2, L/2),
           ylims = (-L/2, L/2),
          xticks = -3:3,
          yticks = -3:3,
          xlabel = "x",
          ylabel = "y",
           title = "a forcing realization",
      framestyle = :box)

Setting initial conditions

Our initial condition is a fluid at rest.

TwoDNavierStokes.set_zeta!(prob, zeros(grid.nx, grid.ny))

Diagnostics

Create Diagnostics; the diagnostics are aimed to probe the energy budget.

E  = Diagnostic(energy,                prob, nsteps=nsteps) # energy
Z  = Diagnostic(enstrophy,             prob, nsteps=nsteps) # enstrophy
diags = [E, Z] # a list of Diagnostics passed to `stepforward!` will  be updated every timestep.

Visualizing the simulation

We initialize a plot with the vorticity field and the time-series of energy and enstrophy diagnostics. To plot energy and enstrophy on the same axes we scale enstrophy with $k_f^2$.

p1 = heatmap(x, y, vars.zeta',
         aspectratio = 1,
                   c = :balance,
                clim = (-40, 40),
               xlims = (-L/2, L/2),
               ylims = (-L/2, L/2),
              xticks = -3:3,
              yticks = -3:3,
              xlabel = "x",
              ylabel = "y",
               title = "vorticity, t=" * @sprintf("%.2f", clock.t),
          framestyle = :box)

p2 = plot(2, # this means "a plot with two series"
               label = ["energy E(t)" "enstrophy Z(t) / k_f²"],
              legend = :right,
           linewidth = 2,
               alpha = 0.7,
              xlabel = "μ t",
               xlims = (0, 1.1 * μ * nsteps * dt),
               ylims = (0, 0.55))

l = @layout Plots.grid(1, 2)
p = plot(p1, p2, layout = l, size = (900, 420))

Time-stepping the Problem forward

Finally, we time-step the Problem forward in time.

startwalltime = time()

anim = @animate for j = 0:round(Int, nsteps / nsubs)
  if j % (1000/nsubs) == 0
    cfl = clock.dt * maximum([maximum(vars.u) / grid.dx, maximum(vars.v) / grid.dy])

    log = @sprintf("step: %04d, t: %d, cfl: %.2f, E: %.4f, Z: %.4f, walltime: %.2f min",
          clock.step, clock.t, cfl, E.data[E.i], Z.data[Z.i], (time()-startwalltime)/60)
    println(log)
  end

  p[1][1][:z] = vars.zeta
  p[1][:title] = "vorticity, μt = " * @sprintf("%.2f", μ * clock.t)
  push!(p[2][1], μ * E.t[E.i], E.data[E.i])
  push!(p[2][2], μ * Z.t[Z.i], Z.data[Z.i] / forcing_wavenumber^2)

  stepforward!(prob, diags, nsubs)
  TwoDNavierStokes.updatevars!(prob)
end

gif(anim, "twodturb_forced.gif", fps=18)

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